Download Aalborg Universitet Force-dependent kinematics: a new analysis method for non-conforming joints

Aalborg Universitet
Force-dependent kinematics: a new analysis method for non-conforming joints
Andersen, Michael Skipper; Damsgaard, Michael; Rasmussen, John
Publication date:
2011
Document Version
Accepted manuscript, peer reviewed version
Link to publication from Aalborg University
Citation for published version (APA):
Andersen, M. S., Damsgaard, M., & Rasmussen, J. (2011). Force-dependent kinematics: a new analysis method
for non-conforming joints. Abstract from 13th Biennial International Symposium on Computer Simulation in
Biomechanics, Leuven, Belgium.
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XIII International Symposium on Computer Simulation in Biomechanics
June 30th - July 2nd 2011, Leuven, Belgium
FORCE-DEPENDENT KINEMATICS: A NEW ANALYSIS METHOD FOR NONCONFORMING JOINTS
Michael Skipper Andersen 1, Michael Damsgaard 2 and John Rasmussen 1
1
M-tech, The AnyBody Research Project, Aalborg University, Denmark.
2
AnyBody Technology A/S, Aalborg, Denmark.
Email: [email protected]
INTRODUCTION
In traditional inverse dynamic analysis of musculoskeletal models [1,2], kinematics and kinetics are treated separately. Kinematic analysis is
performed to compute the position, velocity and
accelerations of the segments in the model from
a description of the joints and the motion. This
is formulated through the solution of a set of
nonlinear constraint equations. Subsequently,
the computed segment positions, velocities and
accelerations are substituted into the dynamic
equilibrium equations to obtain a set of equations with only unknown reaction and muscle
forces. This set of equations is solved through
muscle recruitment by assuming a criterion for
the distribution of the internal forces between
the muscles and the reactions. This approach
requires that the joints and motion of the model
can be completely described through these
constraint equations without consideration of
the forces that created the motion. However,
several anatomical and prosthetic joints, such as
spinal disks, knees and many shoulders are nonconforming to such an extent that the forces significantly influence the detailed joint kinematics
and the joint’s internal force equilibrium. For instance, in the knee, the internal motions occurring
are due to a complex interaction between the
muscle actions, cartilage contact mechanics, and
soft tissue stiffness and deformations. Forward
dynamics approaches have the potential to address these phenomena, but they do not handle
complex muscle systems well and often require
careful tuning and long computation times to
simulate vibrations and damping which in many
cases are not very relevant to orthopaedic problems.
To address this problem, we recently developed a
new method called Force-Dependent Kinematics
(FDK) [3]. The method is used to compute the
muscle and reaction forces as well as notpredetermined motion in user-specified directions.
In this study, we applied FDK to a simple test
problem to model the elbow joint in a 2D model of
the arm using contact forces based on two points
Fig 1: 2D forearm and upper arm model lifting a dumbbell weighing 5 kg. The elbow is modeled using FDK.
in contact with the surface of a cylinder as well as
linear springs, mimicking bone contacts as well as
ligaments, respectively (Fig. 1).
METHODS
The FDK method is based on an assumption of
quasi-static force equilibrium between all the acting forces in the model in the directions in which it
is desired to also compute the displacements
(denoted ). Computationally,
is computed by
introducing a kinematic driver equation (
) in a standard inverse dynamic analysis
model to obtain a kinematically determinate system for a given . The function
of the
model coordinates, , and time, , computes the
FDK directions. We assume that the time derivatives of
are zero, i.e. we obtain a quasi-static
solution. FDK reaction forces, , in the same
directions are also introduced. Hereby, the muscle and reaction forces, including the FDK reaction forces, required to balance the model for a
given
can be computed. The desired FDK displacements bring all the forces in the model in
balance, i.e. where the FDK reaction forces are
zero. This leads to the computational workflow
illustrated in Fig. 2.
A simple 2D test model was constructed. The
model consisted of the forearm, upperarm, elbow joint, a dumbbell, weighing 5 kg, and seven
Fig 2: FDK analysis framework. The displacements in the
FDK directions ( s ) are computed as the displacements,
where no artificial reactions (Fs ) are required to balance
the model in these directions.
muscle units. The muscles were assumed to be
equally strong and with constant strength of 300
N. Three elbow models were tested; a revolute
joint and two models with linear contact forces
and five linear springs, mimicking bone contacts
and ligament actions around the elbow. Contact
forces were computed between two points attached to the upper arm (the blue points in Fig.
1) and a cylinder attached to the forearm. The
five springs were equally spaced on a circle in
the upper arm and they all attached to the same
point in the forearm. This point was also used
as the center of rotation in the revolute joint.
The model performed a one second lift of the
dumbbell by flexing the elbow and shoulder 45O
and 30O, respectively, starting with the elbow
flexed 90O and the shoulder extended 10O.
RESULTS AND DISCUSSION
When the stiffness in the elbow model was
high, the model practically produced the same
results as with a revolute joint, i.e. no joint
translations and the same muscle forces. However, when the stiffness of both the contact as
well as the springs was lowered, the results
were significantly different. The detailed motion
of the joint was significantly different between
the two joint models with high and low stiffness
(Fig. 3). The model with high stiffness showed
practically no joint translations, similarly to a
revolute joint. With low stiffness, however, the
simulation results showed translations of up to
Fig 3: Computed joint translations in the elbow joint.
Notice that with high stiffness, the translations are practically zero, whereas with low stiffness, there are some
translations in the joint.
Fig 4: A TKR model is just one possible practical application of the FDK method.
8 mm.
Although the simple arm example illustrated
here is not physiological, it demonstrates nicely
the features of the FDK method of simultaneously computing the muscle and joint reaction forces, while still allowing modeling of the
joints in terms of the force elements. Applications of the method on real problems are under
development. For instance, a total knee replacement (TKR) model, an anatomical knee
model and modeling of spinal fixation devices
are under development. The TKR model is illustrated in Fig. 4 and preliminary results are presented in [3]. The detailed knee mechanics of
the TKR model is modeled by the contacts between the implant components and the ligaments.
CONCLUSIONS
While we only demonstrated the FDK methodology on a simple 2D model, the same method
works for more complicated 3D models as well.
Due to the FDK method’s capability to compute
both the model motion and internal forces
through quasi-static force equilibrium, it opens
up new possibilities in detailed modeling of
joints in musculoskeletal models. This can be
accomplished by creating models directly of the
mechanical structures of the joint in terms of
force elements, rather than converting the actions of the force elements into kinematic constraint equations. This enables analysis of nonconforming joints under the influence of muscle
forces.
REFERENCES
1. Klein Horsman MD, et al., Clin. Biomechanics,
22(2): 239-247- 2007.
2. Erdemir A, et al., Clin. Biomechanics, 22(2):
131-154, 2007.
3. Andersen MS, et al., Proceedings of ISB.
2011. (Accepted).